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We introduce a method to improve the tractability of the well-known Sample
Average Approximation (SAA) without compromising important theoretical
properties, such as convergence in probability and the consistency of an
independent and identically distributed (iid) sample. We consider each scenario
as a polyhedron of the mix of first-stage and second-stage decision variables.
According to John's theorem, the Lowner-John ellipsoid of each polyhedron will
be unique which means that different scenarios will have correspondingly
different Lowner-John ellipsoids. By optimizing the objective function
regarding both feasible regions of the polyhedron and its unique Lowner-John
ellipsoid, respectively, we obtain a pair of optimal values, which would be a
coordinate on a two-dimensional plane. The scenarios, whose coordinates are
close enough on the plane, will be treated as one scenario; thus our method
reduces the sample size of an iid sample considerably. Instead of using a large
iid sample d查看全文